Example of two different potentials which have practically the same fixed-energy phase shifts

TL;DR

This paper demonstrates that the Newton-Sabatier inversion method for fixed-energy phase shifts is unreliable and can produce different potentials with similar phase shifts, highlighting limitations in potential reconstruction.

Contribution

It reveals the limitations of the Newton-Sabatier method and provides a numerical approach to find different potentials with similar phase shifts.

Findings

Newton-Sabatier procedure is a parameter-fitting method, not a true inversion.

The method cannot guarantee accurate potential reconstruction.

Numerical examples show different potentials can produce similar phase shifts.

Abstract

It is shown that the Newton-Sabatier procedure for inverting the fixed-energy phase shifts for a potential is not an inversion method but a parameter-fitting procedure. Theoretically there is no guarantee that this procedure is applicable to the given set of the phase shifts, if it is applicable, there is no guaran- tee that the potential it produces generates the phase shifts from which it was reconstructed. Moreover, no generic potential, specifically, no potential which is not analytic in a neighborhood of the positive real semiaxis can be reconstructed by the Newton-Sabatier procedure. A numerical method is given for finding spherically symmetric compactly supported potentials which produce practically the same set of fixed-energy phase shifts for all values of angular momentum. Concrete example of such potentials is given.